Problem 3 - Calculus Single Variable

Question

Describes the position of an object at time \(t .\) Calculate the instantaneous velocity at time \(c\). \(p(t)=-7 t^{2} \quad c=3\)

Step-by-Step Solution

Answer

The instantaneous velocity at \( t = 3 \) is \( -42 \).

Step 1: Understand the Formula for Instantaneous Velocity

The instantaneous velocity is defined as the derivative of the position function with respect to time. This means we need to find \( p'(t) \), which is the derivative of the given function \( p(t) = -7t^2 \).

Step 2: Differentiate the Position Function

To find \( p'(t) \), apply the power rule of differentiation: \( \frac{d}{dt}[t^n] = nt^{n-1} \). Thus, for \( p(t) = -7t^2 \), the derivative becomes \( p'(t) = -14t \).

Step 3: Substitute c into the Derivative

To find the instantaneous velocity at \( t = c \), substitute \( c = 3 \) into \( p'(t) = -14t \). This gives \( p'(3) = -14 \times 3 = -42 \).

Key Concepts

Derivative

In calculus, the derivative is a fundamental concept that represents the rate at which a function is changing at any given point. When we talk about derivatives in the context of position and velocity, it refers to how the position of an object is changing over time. The derivative gives us the exact rate of change of the function at a specific point, allowing us to understand the instantaneous characteristics of the function.

• The derivative of a function, denoted as \( f'(x) \), describes how a small change in \( x \) affects the change in the function \( f(x) \).
• It is the slope of the tangent line to the curve at a particular point.
• For our specific exercise, \( p(t) = -7t^2 \), the derivative \( p'(t) \) tells us how the position changes with respect to time.

To compute the derivative, we utilize differentiation techniques, one of the most straightforward being the power rule, which simplifies the process of finding how these changes occur as \( t \) varies.

Instantaneous Velocity

Instantaneous velocity is the speed and direction of an object at a specific instant in time, often perceived as a snapshot of its motion. In contrast to average velocity, which considers the total displacement over time, instantaneous velocity captures the motion at a precise moment by utilizing derivatives.

• Determining the instantaneous velocity involves differentiating the position function with respect to time \( t \).
• In our example, we use the derived function \( p'(t) = -14t \) to calculate the velocity at any specific time.
• Substituting the given \( t = c \) into the derivative helps us find the exact velocity at that moment.

When \( c = 3 \), by plugging into \( p'(t) \), we get \( p'(3) = -42 \). This value represents the instantaneous velocity at \( t = 3 \) seconds, indicating a velocity of \( -42 \) units per given time scale, emphasizing the object's speed and direction at that precise time.

Power Rule

The power rule is a quick and handy tool in calculus for finding the derivative of functions that are represented as a power of \( t \), which in this exercise is vital for easily solving for instantaneous velocity. It follows the rule: if you have a term \( t^n \), its derivative is \( nt^{n-1} \).

• It simplifies the differentiation process, focusing on reducing both time and errors.
• In our task, the position function is \( p(t) = -7t^2 \). Applying the power rule here means multiplying the power by the coefficient, resulting in \( -14t^{2-1} = -14t \).
• This straightforward rule is helpful for derivatives involving polynomial expressions, providing a quick route to understanding the dynamics of functions.

Differentiation using the power rule in such exercises emphasizes the simplicity and elegance of calculus, allowing you to seamlessly transition from the position of an object to its instantaneous velocity. This rule serves as a fundamental stepping stone for any beginner venturing into calculus, making the daunting task of differentiation accessible and manageable.